Question

Carolyn and Sanjay are neighbors. Each owns a car
valued at $10,000. Neither has comprehensive insurance
(which covers losses due to theft). Carolyn’s
wealth, including the value of her car is $80,000.
Sanjay’s wealth, including the value of his car is
$20,000. Carolyn and Sanjay have identical utility
of wealth functions, U(W) = W0.4. Carolyn and
Sanjay can park their cars on the street or rent space
in a garage. In their neighborhood, a street-parked
car has a 50% probability of being stolen during the
year. A garage-parked car will not be stolen.
a. What is the largest amount that Carolyn is willing
to pay to park her car in a garage? What is the
maximum amount that Sanjay is willing to pay?
b. Compare Carolyn’s willingness-to-pay to Sanjay’s.
Why do they differ? Include a comparison
of their Arrow-Pratt measures of risk aversion.
(Hint: See Solved Problem 16.4.) M

Answer 1

a)

Probability of car being stolen=p=0.5

Probability of car not being stolen=1-p=0.5

First we estimate the maximum willingness to pay in case of Carolyn,

Utility in case car is stolen=U(80000-10000=70000)=70000^0.4=86.704016 utils

Utility in case car is not stolen=U(80000)=80000^0.4=91.461010 utils

Expected Utility=p*U(70000)+(1-p)*U(80000)=0.5*86.704016+0.5*91.461010=89.082513 utils

Let Carolyn will pay a maximum sum of X towards for parking. In this case, utility should be at least equal to expected utility i.e.

U(80000-X)=89.082513

(80000-X)^0.4=89.082513

80000-X=74899.88

X=80000-74899.88=$5100.12

Now we estimate the maximum willingness to pay in case of Sanjay,

Utility in case car is stolen=U(20000-10000=10000)=10000^0.4=39.810717 utils

Utility in case car is not stolen=U(20000)=20000^0.4=52.530556 utils

Expected Utility=p*U(10000)+(1-p)*U(20000)=0.5*39.810717+0.5*52.530556=46.170637 utils

Let Sanjay will pay a maximum sum of Y towards for parking. In this case, utility should be at least equal to expected utility i.e.

U(20000-Y)=46.170637

(20000-Y)^0.4=46.170637

20000-Y=14484.87

Y=20000-14484.87=$5515.13

b)

We find that maximum willingness to pay for parking is lower in case of Carolyn.

We can see that marginal utility is decreasing in case of given utility function. it shows that Marginal utility will decrease as income increases.

Initial wealth is higher in case of Carolyn. So, he will pay less to avoid risk.

We are given U(w)=w^0.4

U'(w)=0.4*w^-0.6

U"(w)=-0.24*w^-1.6

Let us find Arrow-Pratt measures of risk aversion

Absolute risk-aversion (ARR)=-u"(w)/u'(w)=-(-0.24*w^-1.6)/(0.4*w^-0.6)=0.6/w

ARR in case of Carolyn=0.6/80000=0.0000075

ARR in case of Sanjay=0.6/20000=0.00003

We can see that ARR is higher in case of Sanjay. It shows that Sanjay will be wiling to pay more to avoid risk.